[Paper Review] Long range scattering for nonlinear Schrödinger equations with critical homogeneous nonlinearity

Satoshi Masaki, Hayato Miyazaki|arXiv (Cornell University)|Dec 14, 2016

Advanced Mathematical Physics Problems7 references4 citations

TL;DR

This paper establishes a sufficient condition for the existence of a modified wave operator in the long-range scattering theory of nonlinear Schrödinger equations with critical homogeneous nonlinearity in one and two spatial dimensions. By decomposing the nonlinearity via Fourier series into resonant and non-resonant parts, the authors derive decay conditions on the Fourier coefficients that ensure convergence of the non-resonant series, thereby enabling the construction of modified wave operators for non-polynomial nonlinearities.

ABSTRACT

In this paper, we consider the final state problem for the nonlinear Schrödinger equation with a homogeneous nonlinearity which is of the long range critical order and is not necessarily a polynomial, in one and two space dimensions. As the nonlinearity is the critical order, the possible asymptotic behavior depends on the shape of the nonlinearity. The aim here is to give a sufficient condition on the nonlinearity to construct a modified wave operator. To deal with a non-polynomial nonlinearity, we decompose it into a resonant part and a non-resonant part via the Fourier series expansion. Our sufficient condition is then given in terms of the Fourier coefficients. In particular, we need to pay attention to the decay of the Fourier coefficients since the non-resonant part is an infinite sum in general.

Motivation & Objective

• To determine the asymptotic behavior of small, low-frequency solutions to the nonlinear Schrödinger equation with critical homogeneous nonlinearity in 1D and 2D.
• To extend the theory of long-range scattering beyond polynomial nonlinearities to general homogeneous nonlinearities of critical order.
• To establish a sufficient condition on the nonlinearity—expressed in terms of Fourier coefficients—that ensures the existence of a modified wave operator.
• To address the challenge of handling non-resonant parts as infinite series in the non-polynomial case, particularly when decay of Fourier coefficients is critical for convergence.

Proposed method

• Represent the homogeneous nonlinearity $ F(u) $ via a periodic function $ g(\theta) $ on the unit circle, exploiting homogeneity to reduce the problem to angular dependence.
• Expand $ g(\theta) $ in a Fourier series to decompose $ F(u) $ into a resonant part (zero-frequency term) and a non-resonant part (higher harmonics).
• Analyze the non-resonant part as an infinite sum of terms involving $ |u|^{2m+1}u $, with coefficients derived from the Fourier expansion.
• Establish $ L^2 $-type estimates for the non-resonant terms using decay properties of the Fourier coefficients $ g_n $, particularly requiring $ |g_n| \lesssim |n|^{-\beta} $ for $ \beta > 0 $.
• Use Strichartz and smoothing estimates in a weighted function space $ X_{d,T,b} $ to control the solution norm and prove contraction in a fixed-point argument.
• Prove Lipschitz continuity of the nonlinearity in terms of the Fourier coefficients, ensuring the well-posedness of the integral formulation.

Experimental results

Research questions

• RQ1 Under what conditions on the nonlinearity does the nonlinear Schrödinger equation admit a modified wave operator in the long-range scattering regime?
• RQ2 How can one handle the non-resonant part of a non-polynomial, homogeneous nonlinearity in the critical case, especially when it forms an infinite series?
• RQ3 What decay rate on the Fourier coefficients of the nonlinearity ensures convergence and existence of the modified wave operator?
• RQ4 Can the asymptotic behavior of solutions be characterized when the nonlinearity is not a polynomial, such as $ F(u) = |\operatorname{Re} u| \operatorname{Re} u $ in 2D?
• RQ5 How does the presence of a resonant term (governed by $ g_0 $) affect the long-range behavior compared to the non-resonant terms?

Key findings

• A sufficient condition for the existence of a modified wave operator is derived in terms of the Fourier coefficients $ g_n $ of the nonlinearity: if $ |g_n| \lesssim |n|^{-\beta} $ for some $ \beta > 0 $, the non-resonant series converges in the relevant function space.
• The nonlinearity $ F(u) = |\operatorname{Re} u| \operatorname{Re} u $ in 2D is shown to be long-range, as its Fourier expansion contains a non-zero resonant term and the non-resonant part converges due to sufficient decay of coefficients.
• The solution constructed via the fixed-point argument lies in the space $ X_{d,T,b} $, with norm estimates showing $ \|\Phi(v)\|_{X_{d,T,b}} < 1 $ for large $ T $, ensuring existence and uniqueness.
• The contraction mapping argument in $ X_{d,T,b} $ relies on smallness of $ \|g\|_{\text{Lip}} \varepsilon^{2/d} $, which is ensured by choosing $ \varepsilon $ small enough depending on the Lipschitz constant of $ g $.
• The asymptotic behavior of the solution is shown to be of the form $ u(t) \sim (2it)^{-d/2} e^{i|x|^2/4t} \widehat{u_+}(x/(2t)) \exp(-i \mu |\widehat{u_+}(x/(2t))|^{2/d} \log t) $, confirming long-range scattering when the resonant term is non-zero.
• The paper establishes that the Lipschitz continuity of the function $ g(\theta) $, which parametrizes the nonlinearity, is equivalent to the pointwise bound $ |F(u) - F(v)| \leq C(|u|^{2/d} + |v|^{2/d})|u - v| $, crucial for the contraction argument.